Rich arc routing problem in city logistics: Models and solution algorithms using a fluid queue-based time-dependent travel time representation

• Systematically consider time-varying traffic conditions and system-wide impact in urban city logistic problem. • Incorporate parsimonious road link travel time functions with First-In-First-Out property. • Develop three unified optimization models for characterizing problem-specific features in real-life rich arc routing applications. • Exact solution algorithms using Lagrangian relaxation and branch-and-price methods. City logistics, as an essential component of the city operation system, aims at managing the complex flow of goods and services from providers to customers efficiently. Delays associated with peak-period traffic congestion exists in both large and small metropolitan areas. As many of the service tasks in city logistics are needed to be performed during peak hours, operators of urban management movement should consider reducing the total trip time and delay when designing service plans. Equally important, the congestion impact of service vehicles to other road users should also be considered. In this paper, we focus on formulating and solving rich arc routing problems (RARPs) in city logistics with a congested urban environment. We highlight the needs of embedding a structurally parsimonious time-dependent travel time model in RARP for producing high-quality and practically useful solutions. A fluid queue model based analytical approach is presented for link travel time calibration in the form of polynomial arrival rate functions. Accordingly, system-wide (societal) impact of vehicles routing is analytically derived and incorporated into the RARP models which enables traffic managers to systematically consider operation costs and societal impacts when designing routing policies in real-life city logistics applications. Additionally, we develop two new representation schemes for time-dependent travel time modeling in RARPs, including a discretized time-expanded representation scheme and a nonlinear polynomial representation scheme. Three modeling approaches for RARPs are proposed, with different perspectives of capturing time-dependent travel time and formulating problem-specific constraints. With a real-life sprinkler truck routing problem as the representative example of RARP, we develop two efficient exact solution algorithms, including a Lagrangian relaxation-based method and a branch-and-price based method. The latter one is embedded with an enhanced parallel branch-and-bound algorithm. Extensive numerical experiments are conducted based on real-world networks and traffic flow data to demonstrate the effectiveness of the proposed methods.